3.34 \(\int \cos ^3(x) \sin (x) \, dx\)

Optimal. Leaf size=8 \[ -\frac{1}{4} \cos ^4(x) \]

[Out]

-Cos[x]^4/4

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Rubi [A]  time = 0.0282481, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ -\frac{1}{4} \cos ^4(x) \]

Antiderivative was successfully verified.

[In]  Int[Cos[x]^3*Sin[x],x]

[Out]

-Cos[x]^4/4

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Rubi in Sympy [A]  time = 1.12211, size = 7, normalized size = 0.88 \[ - \frac{\cos ^{4}{\left (x \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(cos(x)**3*sin(x),x)

[Out]

-cos(x)**4/4

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Mathematica [A]  time = 0.00187126, size = 8, normalized size = 1. \[ -\frac{1}{4} \cos ^4(x) \]

Antiderivative was successfully verified.

[In]  Integrate[Cos[x]^3*Sin[x],x]

[Out]

-Cos[x]^4/4

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Maple [A]  time = 0.004, size = 7, normalized size = 0.9 \[ -{\frac{ \left ( \cos \left ( x \right ) \right ) ^{4}}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cos(x)^3*sin(x),x)

[Out]

-1/4*cos(x)^4

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Maxima [A]  time = 1.35486, size = 8, normalized size = 1. \[ -\frac{1}{4} \, \cos \left (x\right )^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)^3*sin(x),x, algorithm="maxima")

[Out]

-1/4*cos(x)^4

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Fricas [A]  time = 0.213085, size = 8, normalized size = 1. \[ -\frac{1}{4} \, \cos \left (x\right )^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)^3*sin(x),x, algorithm="fricas")

[Out]

-1/4*cos(x)^4

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Sympy [A]  time = 0.037774, size = 7, normalized size = 0.88 \[ - \frac{\cos ^{4}{\left (x \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)**3*sin(x),x)

[Out]

-cos(x)**4/4

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GIAC/XCAS [A]  time = 0.200644, size = 8, normalized size = 1. \[ -\frac{1}{4} \, \cos \left (x\right )^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)^3*sin(x),x, algorithm="giac")

[Out]

-1/4*cos(x)^4